In the above image see the two seatings, in both the seatings everyone have same neighbour.
So when calculating the permutations consider them as same and not different.
total no of ways six people around a circle table is 5!+120
but if left right neighbor of a people is same as right left neighbor for all 6 people then they we be consider as same means no we get should be divided by 2 (see the image in comment by sakharam )
so answer is 120/2=60
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